局部对称伪Riemann流形中的紧致极大类时子流形

李 影,宋卫东

(安徽师范大学 数学计算机科学学院,安徽 芜湖 241003)



研究简报

局部对称伪Riemann流形中的紧致极大类时子流形

李 影,宋卫东

(安徽师范大学 数学计算机科学学院,安徽 芜湖 241003)

利用活动标架法,得到了局部对称伪Riemann流形中极大类时子流形的一个Simons型积分不等式,以及该子流形成为全测地类时子流形的关于其第二基本形式模长平方的拼挤定理.

伪Riemann流形;局部对称;极大类时子流形;全测地类时子流形

本文约定各类指标取值范围如下:

1≤A,B,C,…≤n+p; 1≤i,j,k,…≤n;n+1≤α,β,γ,…≤n+p.

(1)

(2)

则有

(3)

(4)

类似地,曲率张量场Kαijk的共变导数Kαijk,l定义为

限制到Mn上时,有

(5)

(1-δ),A≠B;

(1-δ),A,B,C,D互不相同.

(6)

再由式(1),(2),(5),(6)得

(7)

(8)

(9)

(10)

由于(tr(HαHβ))p×p是实对称矩阵,因此选取法标架场{eα}可使之对角化,即

(11)

从而有

(12)

从而

(13)

由文献[10]，显然有

(14)

由式(7)～(14),有

(15)

由于Mn是紧致无边的,根据Stocks定理,对式(15)两边积分得

证毕.

(16)

证明:由已知条件式(16)可知式(15)的右边非负,而Mn是紧致无边的,由Hopf极大值原理可知S为常数.从而式(15)左边为零.因此,式(15)右边也为零,即

结合已知条件可知S=0,故Mn是全测地类时子流形.证毕.

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(责任编辑：赵立芹)

MaximumTimelikeSubmanifoldinaLocallySymmetricPseudo-RiemannianManifold

LI Ying,SONG Weidong

(CollegeofMathematicsandComputerScience,AnhuiNormalUniversity,Wuhu241003,AnhuiProvince,China)

Based on the moving frames,an integral inequality about maximal timelike submanifold was obtained in the locally symmetric pseudo-Riemannian manifold and a pinching theorem about the squared norm of the second fundamental form for the compact maximal timelike submanifold was gived in locally symmetric pseudo-Riemannian manifold.

pseudo-Riemannian manifold;locally symmetric;maximum timelike submanifold;totally geodesic timelike submanifold

10.13413/j.cnki.jdxblxb.2015.03.20

2014-09-15.

李 影(1991—),女,汉族,硕士研究生,从事微分几何的研究,E-mail:909789714@qq.com.通信作者:宋卫东(1958—),男,汉族,教授,从事微分几何的研究,E-mail:swd56@sina.com.

国家自然科学基金(批准号:11071005)和安徽省教育厅自然科学重点项目(批准号:KJ2010A125).

O186.12

：A

：1671-5489(2015)03-0457-04